Interpretations
What does actually mean in the real world?
Introduction
Just like tells you how fast changes, tells you how fast changes if you move East (along x-axis), and tells you how fast changes if you move North (along y-axis).
Visualizing Rates of Change
Interactive: The Saddle Point
At this point, moving in (Red) drops you down, but moving in (Green) pushes you up.
Tangent Lines
Vector Equations
The tangent lines can be written as vector functions:
Tangent in x-direction
Direction vector:
Tangent in y-direction
Direction vector:
Worked Examples
Example 1: Rate of Change Calculation
Let . Find the rate of change in at .
1. Find .
2. Plug in : .
The function is increasing at a rate of 4 units per unit x.
Example 2: Tangent Line Equation
Find the tangent line to at in the -direction.
1. Evaluate function: . Point: .
2. Find slope: .
3. Direction vector for y-tangent: .
4. Line Eq: .
Example 3: Steepness Comparison
For at , which is steeper: the slope in the direction or direction?
1. Calculate :
.
2. Calculate :
.
3. Compare:
Slope in x (48) is much larger than slope in y (16).
The surface is steeper in the x-direction.
Practice Quiz
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